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Sparse approximation (also known as sparse representation) theory deals with sparse solutions for systems of linear equations. Techniques for finding these solutions and exploiting them in applications have found wide use in image processing , signal processing , machine learning , medical imaging , and more.
Sparse dictionary learning (also known as sparse coding or SDL) is a representation learning method which aims to find a sparse representation of the input data in the form of a linear combination of basic elements as well as those basic elements themselves.
In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. [1] There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as sparse but a common criterion is that the number of non-zero elements is roughly equal to the number of ...
In applied mathematics, k-SVD is a dictionary learning algorithm for creating a dictionary for sparse representations, via a singular value decomposition approach. k-SVD is a generalization of the k-means clustering method, and it works by iteratively alternating between sparse coding the input data based on the current dictionary, and updating the atoms in the dictionary to better fit the data.
Similarly, such a constraint can be applied to its representation itself, generating a cascade of sparse representations: Each code is defined by a few atoms of a given set of convolutional dictionaries. Based on these criteria, yet another extension denominated multi-layer convolutional sparse coding (ML-CSC) is proposed.
If contains a large number of vectors, searching for the most sparse representation of is computationally unacceptable for practical applications. In 1993, Mallat and Zhang [ 1 ] proposed a greedy solution that they named "Matching Pursuit."
This concept was reintroduced by David Donoho and Michael Elad in the context of sparse representations. [5] A special case of this definition for the two-ortho case appeared earlier in the paper by Donoho and Huo. [6] The mutual coherence has since been used extensively in the field of sparse representations of signals.
Shearlets are to date the only directional representation system that provides sparse approximation of anisotropic features while providing a unified treatment of the continuum and digital realm that allows faithful implementation. Extensions of shearlet systems to (), are also available.